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Unformatted text preview: tile 85 (0.185) median 328 (0.63) 0.95fractile 775 (0.185) CHAPTER 10. USING DATA Using Data to Fit Theoretical Probability Models Slide No. 21 ENCE 627 ©Assakkaf Method A: One way to deal with data is simply to fit a theoretical distribution to it. Step 1: Decide what kind of distribution is appropriate (binomial, Poisson, normal, and so on) • What distribution is best? Need to understand the setting – Defects maybe Poisson – Value in [0,1] maybe beta – Normal? Need symmetry as well as other things 11 Slide No. 22 CHAPTER 10. USING DATA Using Data to Fit Theoretical Probability Models ENCE 627 ©Assakkaf Step 2: Choose the values of the distribution parameters • Having chosen the distribution, need to calibrate, i.e., choose the values for the parameters. Bernoulli (P), Binomial (n, p), Poisson ( λ), etc. • Easy way (probably adequate in a number of settings. • Take sample mean and sample variance: X= Statistical reasons Statistical reasons why not n why not n 1n ∑ xi n i =1 S2 = 1n ∑ ( xi − X ) 2 n − 1 i =1 Slide No. 23 CHAPTER 10. USING DATA Using Data to Fit Theoretical Probability Models ENCE 627 ©Assakkaf Example: Calculate the sample mean (x) and sample variance (S2) for the 35 halfway house observations n = 35 1n X = ∑ xi = 380.4 n i =1 S2 = ( 1n ∑ xi − X n − 1 i= ) 2 = 47,344.3 S = 47,344.3 = 217.6 We might choose a normal distribution with mean µ = 380.4 and standard deviation σ = 217.6 to represent the distribution of the yearly bed-rental costs. 12 CHAPTER 10. USING DATA Using Data to Fit Theoretical Probability Models Slide No. 24 ENCE 627 ©Assakkaf Method B: Fit a theoretical distribution using fractiles. That is, find a theoretical distribution whose fractiles match as well as possible with the fractiles of the empirical data. In this case we would be fitting a theoretical distribution to a data-base distribution. CHAPTER 10. USING DATA Using Data to Fit Theoretical Probability Models Slide No. 25 ENCE 627 ©Assakkaf Method C: For most initial attempts to model uncertainty in a decision analysis, it may be adequate to use the sample mean and variance as estimates of the mean and variance of the theoretical distribution and to establish parameter values in this way. Refinement of the probability model may require more careful judgment about the kind of distribution as well as more care in fitting the parameters. 13 CHAPTER 10. USING DATA Slide No. 26 ENCE 627 ©Assakkaf Testing the Validity of Assumed Distribution When a theoretical distribution has been assumed, the validity of the assumed distribution may be verified or disproved statistically by goodness-of-fit tests. Two tests are commonly used: – The Chi-square – The Kolmogorov-Smirnov test CHAPTER 10. USING DATA Slide No. 27 ENCE 627 ©Assakkaf Testing the Validity of Assumed Distribution Chi-square Test for Goodness of Fit – Consider a sample of O observed values of a random variable. – The chi-square goodness-of-fit test compares the observed frequencies O1, O2,…, Ok of k values (k intervals) of the variate with the corresponding frequencies E1, E2,…,Ek...
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This note was uploaded on 10/24/2012 for the course ESI 6385 taught by Professor Mansoorehmollaghasemi during the Fall '12 term at University of Central Florida.

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